Defines and evaluates denominators in the RG equations. The denominators in the RG equations are $$ d_0 = \omega - \frac{1}{2}\left(D - \mu\right) - \frac{U}{2} + \frac{K}{2} $$ $$ d_1 = \omega - \frac{1}{2}\left(D - \mu\right) + \frac{U}{2} + \frac{J}{2} $$ $$ d_2 = \omega - \frac{1}{2}\left(D - \mu\right) + \frac{J}{4} + \frac{K}{4} $$

RG Equations

The RG equations for the symmetric spin-charge Anderson-Kondo are

$$ \Delta U = 4|V|^2 \left[\frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) + \frac{U}{2} + \frac{1}{2}J} - \frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) - \frac{U}{2} + \frac{1}{2}K}\right] + \sum_{k<\Lambda_j} \frac{3}{4}\frac{K^2 - J^2}{\omega - \frac{1}{2}\left(D - \mu\right) + \frac{1}{4}J + \frac{1}{4}K} $$$$ \Delta V = \frac{V K}{16}\left(\frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) - \frac{U}{2} + \frac{1}{2}K} + \frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) + \frac{1}{4}J + \frac{1}{4}K} \right) - \frac{3VJ}{4}\left( \frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) + \frac{U}{2} + \frac{1}{2}J} + \frac{1}{\omega - \frac{1}{2}\left(D - \mu\right) + \frac{1}{4}J + \frac{1}{4}K} \right) $$$$ \Delta J = - J^2\left(\omega - \frac{1}{2}\left(D - \mu\right) + \frac{1}{4}J + \frac{1}{4}K\right)^{-1} $$$$ \Delta K = - K^2\left(\omega - \frac{1}{2}\left(D - \mu\right) + \frac{1}{4}J + \frac{1}{4}K\right)^{-1} $$

The following equation accepts the coupling values at the $j^{th}$ step of the RG, applies the RG equations on them and returns the couplings for the $(j-1)^{th}$ step. If any coupling changes sign, it is set to 0.

The following function does one complete RG for a given set of bare couplings and returns arrays of the flowing couplings.

1. $V=0$

First we will look at the simplified case of $V=0$. Since the RG equation for $V$ involves $V$, it will not flow. We need to look only at $U$, $J$ and $K$. Depending on the value of $\omega$, the denominator can be either positive or negative. We look at the two cases separately.

a. $\omega - \frac{\epsilon_q}{2} + \frac{1}{4}J + \frac{1}{4}K>0$ (high $\omega$):

These aren't truly URG fixed points because the denominator will not converge towards zero.

i. $J=K$

Since $\Delta U \propto K^2 - J^2$, $U$ will be marginal here.

ii. $J > K$

Since $\Delta U \propto K^2 - J^2$, $U$ will be irrelevant here.

iii. $J < K$

Since $\Delta U \propto K^2 - J^2$, $U$ will be relevant here.

b. $\omega - \frac{\epsilon_q}{2} + \frac{1}{4}J + \frac{1}{4}K<0$ (low $\omega$):

This is the regime where we achieve true strong-coupling fixed points in $J,K$. The signature of $K^2 - J^2$ will determine whether $U$ is relevant or irrelevant.

i. $J>K$

i. $J<K$

To wrap up the $V=0$ case, we look at an RG-invariant:

$\frac{\Delta J}{\Delta K} = \frac{J^2}{K^2} \implies \frac{1}{J} - \frac{1}{K} = \frac{1}{J_0} - \frac{1}{K_0}$

Note that this is an invariant even when $V$ is turned on.

Phase Diagram

2. $V > 0$

The inclusion of $V$ will mean that there will not by any sharply defined phase of $U^*$ any more. We will still be working in the regime where $J,K$ flow to strong-coupling, and since those RG equations do not depend on $V$, their flows are unchanged. The behaviour of $U$ will get complicated however. To make sense, we will see how the total (over a range of $\omega$ and bare $U$) number of fixed points where $U^* > U_0$ and the total number of fixed points where $U^* < U_0$, in each of the four quadrants of the phase diagram, varies against the bare value $V_0$.

a. Behaviour of distribution of fixed points as a function of bare $V$

We can classify the fixed points into three classes: $U*=0$, $U^* > U_0$ and $U^* < U_0$. The number of fixed points in each class for $V=0$ has already been clarified in the $V=0$ section, specially in the phase diagram. For that, we will first create some helper functions.

We will first check how the $c_i$ vary as functions of $V$, in each quadrant.

First Quadrant: $U>0, J>K$

As $D$ increases, dominant fixed point switches from $U^*>0$ to $U^*=0$ at some critical $V_c$. The critical V appears to decrease with D initially, but later increases (shown later). For large $D$, this critical V will be inaccessible, and the flip will be forbidden, leading to a phase where $U^*>U_0$.

Second Quadrant: $U>0, J<K$

With increase in $D$, the hump keeps moving forward, and disappears at a sufficiently large $D$, so the dominant behaviour is unchanged (still $U^*=0$).

Third Quadrant: $U<0, J<K$

We see the opposite of the first quadrant behaviour here. There is again a flip at some critical V, critical V initially increases and then decreases. The fixed point phase should be $U^*<U_0$.

Fourth Quadrant: $U<0, J>K$

In the fourth quadrant, the $V$ causes significant changes only at very small $D$.

$\frac{c_0}{c_1}$ at $V_0=0.02$ and $V_c$, both vs $D$, for the $1^\text{st}$ quadrant

The critical $V$ at which the transition from $U^*>U_0$ to $U^*=0$ occurs is a function of $D$ and $J,K$. It increases with increase in $D$, as well as increase in $J$.

Comparison of $J^*$ and $V^*$

Behaviour of $V$

b. Change in the fraction of irrelevant fixed points under increase in $D$

Next we will see how the ratio of number of fixed points in each class varies as we increase the bandwidth, for a particular $V \sim 10$ in the stable region.

c. Change in the critical $V$ under increase in $D$

For the first and third quadrants, there is a critical value of $V$ at which the number of relevant and irrelevant fixed points become equal. We will now see how this value depends on the bandwidth $D$.